Yes, they DO outgrow separation anxiety and the need to be by your side 24/7 -- just ask the parent of any teenager!! ; -)
I would say keep doing what you're doing, as far as bringing them to Sunday School and having non-relative babysitters. I think that for the first few weeks, it's not the worst idea to have either you or your husband stay in Sunday School with them. The other kids and the teachers will soon become friendly faces. Trust your gut as to when it's time to start leaving them alone there. You could leave them alone for the first half of the service and then return to them for the second half, working up to leaving them alone the whole time.
I will give you my own perspective on separation anxiety, which I think can differ from the norm: it's not "bad", it doesn't mean your child is flawed, or that there is anything wrong with your parenting. I think separation anxiety can show that a child has a good attachment to his parents, that he is comfortable at home and feels safe there. (Of course, a child can have those feelings w/out separation anxiety, too.) Also, some children are just more at ease in social situations than others, and I do not believe you can change anyone's "nature" in this regard.
FWIW, my two cents on Pre-K: if he's not ready, he's not ready. In Minnesota, the law for compulsory ages for school attendance is 7-16, so why people feel pressure to send their 4 & 5 year olds (who may or may not be ready) to school is something I do not understand.
My son is 13 months old and we bring him to Sunday School, too. He usually does OK, but he does have his rough days. Two weeks ago when we went to pick him up, he was sitting by the door sobbing. The teachers were trying to console him, and they had ushered the other kids into another room while my son had his meltdown. (So know that you are not alone!)
Also, my son is very skiddish around strangers, and that is A-OK with me! We live in an apartment building, and even if a stranger's voice carries in from the hallway to our apartment, my son can get a worried look on his face. I know he will outgrow it in time.
Good luck.